Finite Group Databasebeta

← #364149

Subgroups — #364149

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34065C211yesmaximal, characteristic
2C2×#1871C212yesmaximal
3#20335C211yesmaximal, characteristic, complemented
4#13852C211yesmaximal, characteristic, complemented
5#11226C212yesmaximal, complemented
6#11075C212yesmaximal, complemented
7#10808C212yesmaximal, complemented
8#10790C212yesmaximal, complemented
9#10732C212yesmaximal, complemented
10C23×M16C2212yes—
11C2×#525C2211yescharacteristic
12C2×C4×#41C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#104C2214yes—
15#2324C414yescomplemented
16#2323C414yescomplemented
17#2322—22no—
18#2321—22no—
19#2297C2211yescharacteristic
20#2296C2211yescharacteristic
21#2293—22no—
22#2290—22no—
23#2280C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4C2212yes—
27#1871—28no—
28C2×#339C2214yes—
29C2×#339C2214yes—
30C2×#339C2214yes—
31#1729—22no—
32#1729—22no—
33#1728C2212yes—
34#1728C2212yes—
35#1441—22no—
36#1439—22no—
37C4×#95C2212yes—
38C22×#41C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41C22×M16—28no—
42C22×M16—24no—
43C22×M16—24no—
44C22×M16—44no—
45C22×M16C2314yes—
46C23×C8—22no—
47#525—22no—
48#525—22no—
49#525D414yes—
50#525Q814yes—
51C2×#123—21no—
52C2×C22⋊Q8C2×C412yes—
53C2×#121C2×C412yes—
54C2×#120—21no—
55C4×#41—44no—
56C2×C4×Q8C2×C412yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C412yes—
60C2×#117—21no—
61C2×#117—44no—
62C2×#117C2×C414yes—
63C2×#117D412yes—
64C2×#117D412yes—
65C2×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4D412yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D412yes—
72C22×C42—21no—
73C22×C42C2314yes—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#104—28no—
77C2×#104—28no—
78C2×#104—44no—
79#391C2×C412yes—
80#390—24no—
81#390C2×C412yes—
82#389—24no—
83#389C2×C414yes—
84#388—24no—
85#388—24no—
86#388—24no—
87#387—24no—
88#386—22no—
89#386—24no—
90#385—22no—
91#385—24no—
92#381—21no—
93#380C2×C414yes—
94#379—21no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—44no—
97C4×C4⋊C4C2×C414yes—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—24no—
102C4×C22⋊C4—24no—
103#376—21no—
104C2×#95—24no—
105C2×#95—22no—
106C2×#95—22no—
107C43—21no—
108#339—28no—
109#339—28no—
110#339—44no—
111#336—22no—
112#336—22no—
113#336—24no—
114C25C2×D411yescharacteristic
115C2×#41—21no—
116C2×#41—21no—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41C2×D412yes—
120C2×#41C2×Q812yes—
121C22×Q8C22×C411yescharacteristic
122C22×D4—21no—
123C22×D4—22no—
124C22×D4C22×C411yescharacteristic
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—21no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4C22×C412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×Q812yes—
137C23×C4#4112yes—
138C23×C4#4112yes—
139C23×C4C2411yescharacteristic
140C2×M16—28no—
141C2×M16—24no—
142C2×M16—24no—
143C2×M16—24no—
144C2×M16—24no—
145C2×M16—24no—
146C2×M16—416no—
147C2×M16—416no—
148C2×M16—44no—
149C2×M16—44no—
150C2×M16#4118yes—
151C22×C8—28no—
152C22×C8—24no—
153C22×C8—44no—
154C22×C8—44no—
155#123—24no—
156#123—24no—
157C22⋊Q8—24no—
158C22⋊Q8C4⋊C418yescomplemented
159#121—24no—
160#121C4⋊C418yescomplemented
161#120—24no—
162#120—24no—
163C4×Q8—44no—
164C4×D4—28no—
165C4×D4—28no—
166C4×D4—44no—
167C4×D4—44no—
168C4×D4—44no—
169#117—21no—
170#117—21no—
171#117—24no—
172#117—24no—
173#117—48no—
174#117—42no—
175#117—42no—
176#117—42no—
177#117—42no—
178#117—44no—
179#117C2×D412yes—
180#117C2×Q812yes—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4C22⋊C414yes—
189C2×C4⋊C4C22⋊C414yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4C22⋊C414yes—
206C2×C22⋊C4C22⋊C414yes—
207C2×C42—21no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—44no—
221C2×C42C22×C412yes—
222C2×C42C22×C412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D412yes—
226#104—416no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—48no—
232#95—44no—
233#95—44no—
234C24—22no—
235C24—22no—
236C24—21no—
237C24—44no—
238C24C2×C22⋊C412yes—
239C24#12112yes—
240C24C22⋊Q812yes—
241C24C22×D411yescharacteristic
242#41—28no—
243#41—28no—
244#41—44no—
245#41—44no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8C2×C4⋊C412yes—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—28no—
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—84no—
259C2×D4C2×C4⋊C412yes—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—28no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—84no—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C414yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C4⋊C414yes—
300C22×C4#11714yes—
301C22×C4#12114yes—
302C22×C4#12112yes—
303C22×C4C22⋊Q814yes—
304C22×C4C22⋊Q812yes—
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×Q811yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4114yes—
312C22×C4H(5,𝔽2)14yes—
313M16—28no—
314M16—28no—
315M16—416no—
316M16—44no—
317M16—44no—
318C2×C8—28no—
319C2×C8—416no—
320C2×C8—44no—
321C2×C8—44no—
322C4⋊C4—22no—
323C4⋊C4—28no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—416no—
330C4⋊C4—84no—
331C4⋊C4C2×C4⋊C414yes—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—28no—
335C22⋊C4—28no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—416no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C42—24no—
345C42—24no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—44no—
353C42—42no—
354C42—42no—
355C42—44no—
356C42—44no—
357C42—44no—
358C42—44no—
359C42—84no—
360C23—24no—
361C23—22no—
362C23—24no—
363C23—24no—
364C23—22no—
365C23—24no—
366C23—24no—
367C23—22no—
368C23—44no—
369C23—44no—
370C23—41no—
371C23—44no—
372C23—41no—
373C23—82no—
374C23—82no—
375C23—84no—
376C23#38014yes—
377C23C22×C22⋊C411yescharacteristic
378C23C2×#12111yescharacteristic
379C23C2×C22⋊Q811yescharacteristic
380C23#53212yes—
381C23#53412yes—
382Q8—24no—
383Q8—42no—
384D4—24no—
385D4—44no—
386D4—44no—
387D4—42no—
388D4—88no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—41no—
411C2×C4—44no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—88no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4#38014yes—
430C2×C4C22×C4⋊C411yescharacteristic
431C2×C4C2×#11711yescharacteristic
432C2×C4C4×#4112yes—
433C2×C4#51812yes—
434C2×C4C2×#12111yescharacteristic
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4#52512yes—
437C2×C4#53412yes—
438C2×C4#54614yes—
439C2×C4#55114yes—
440C2×C4#55314yes—
441C2×C4#55414yes—
442C8—48no—
443C22—22no—
444C22—22no—
445C22—24no—
446C22—22no—
447C22—22no—
448C22—24no—
449C22—24no—
450C22—24no—
451C22—44no—
452C22—44no—
453C22—42no—
454C22—44no—
455C22—84no—
456C22—84no—
457C22—82no—
458C22—82no—
459C22C2×#38011yescharacteristic
460C22#229711yescharacteristic
461C22#230811yescharacteristic
462C22#254312yes—
463C22#256412yes—
464C22#290114yes—
465C4—24no—
466C4—44no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—44no—
471C4—88no—
472C4—82no—
473C4—82no—
474C4#234912yes—
475C4#235312yes—
476C4#253412yes—
477C4#257912yes—
478C2—22no—
479C2—22no—
480C2—44no—
481C2—82no—
482C2#1396011yescharacteristic
483C2#2065512yes—
484C2#2138412yes—
485C2#2177512yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20335C2semidirect
4. #13852C2semidirect
5. #11226C2semidirect
6. #11075C2semidirect
7. #10808C2semidirect
8. #10790C2semidirect
9. #10732C2semidirect
15. #2324C4semidirect
16. #2323C4semidirect
158. C22⋊Q8C4⋊C4semidirect
160. #121C4⋊C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#340652
C2×#18712
#203352
#138522
#112262
#110752
#108082
#107902
#107322